2+2 Decomposition

We now interrupt the development to emphasize that this linear algebra analysis is not restricted to rectilinear coordinates. We shall see that the coordinates relative to which the Maxwell system can be decoupled (and solved) via the method of eigenvectors and eigenvalues are the cartesian, cylindrical, spherical, and other coordinate orthogonal coordinate systems, with time added as the fourth coordinate. For these the spacetime version of the infinitesimal interval (Pythagorean theorem) assumes the familiar form

$$ds^2 =-dt^2+dz^2+dx^2+dy^2 \textrm{cartesian}$$

$$ds^2 =-dt^2+dz^2+dr^2+r^2d\theta^2 \textrm{cylindrical}$$

$$ds^2 =-dt^2+dr^2+r^2(d\theta^2+\sin^2\theta\, d\varphi^2) \textrm{spherical}$$
or more generally

$$ds^2 =g_{AB}(x^C)dx^Adx^B + R^2(x^C)(d\theta^2+\sin^2\theta\, d\varphi^2 )~~~$$

Their conceptual common denominator is that the first two coordinates - the longitudinal coordinates - are orthogonal to the last two - the transverse coordinates. The longitudinal spatial direction is the propagation direction of e.m. radiation, say in the direction of a cylindrical wave guide or the radial direction of a spherical coordinate system. The two spatial transverse directions point along the cross sectional area of that wave guide or the angular directions of the concentric spheres of constant radii.

This two-plus-two decomposition applies not only to the coordinates and their differentials, but also to four-dimensional vector fields tangent to such coordinate surfaces, e.g.,

$$\left[ \begin{array}{c} \phi\\ A_z\\ A_x\\ A_y \end{array} \right... ...array}{c} ~\\ ~ \end{array}\right\} \textrm{transverse components}, \end{array}$$ (662)

the four-vector potential and the charge density-flux four-vector respectively.

We shall decompose the four-vector potential $\left[ \begin{array}{c} \phi\\ \vec A \end{array} \right]$ into three parts. The key finding from this decomposition is that these parts are eigenvectors of the Maxwell wave operator $\mathcal A$ , Eq.(6.46), and that they are identified with the transverse electric (TE), transverse magnetic (TM), and transverse electric-magnetic (TEM) fields of Maxwell theory ⁶⁷.

The eigenvector decomposition takes advantage of the fact that any two-dimensional vector field, be it longitudinal or transverse, can be decomposed uniquely into the gradient of a scalar function and into what amounts to a pure curl vector field in three dimensions. As a consequence, any four-vector such as those in Eq.() has the unique decomposition

$$\left[ \begin{array}{c} \phi\\ A_z\\ A_x\\ A_y \end{array} \right... ...i }_{\begin{array}{c} \textrm{longitudinal}\\ \textrm{gradient} \end{array}} ~.$$ (663)

This 2+2 decomposition establishes a one-to-one correspondence between four-vector fields and the scalar fields $\Phi,\Phi^{TE}$ and $\Psi,\Phi^{TM}$ in the transverse and longitudinal planes respectively.

The existence and uniqueness of this 2+2 decomposition is established in Exercise 6.2.2 on page .

Footnotes

... theory⁶⁷

The justification for these apellations are given on on Pages , , and , repectively.